Accurate Learning of Equivariant Quantum Systems from a Single Ground State

TL;DR

The paper tackles predicting ground-state properties across parameter space in gapped quantum systems by exploiting equivariance under the automorphism group $G$ of the interaction hypergraph. It proposes an approach that treats different regions of a single ground state as distinct training samples, yielding $|\,\mathcal{S}/G\,|$ per-orbit models to predict a $p$-body observable $O(x)=\sum_I \alpha_I(x)O_I$ across the phase, using random Fourier features and LASSO for scalability. The authors prove that, for periodic systems with $|G|=\Theta(n)$, the prediction error tends to zero in the thermodynamic limit, and provide explicit asymptotic bounds for short-range/exponential and power-law decays. Numerical simulations with DMRG on 1D disordered Heisenberg and long-range Ising chains validate the theory, showing shrinking RMS error with system size and accurate retrieval of two-body correlations via classical shadows. The work offers a practical route to dramatically reduce data requirements for characterizing topological/gapped phases and suggests extensions to gapless regimes and Coulomb interactions.

Abstract

Predicting properties across system parameters is an important task in quantum physics, with applications ranging from molecular dynamics to variational quantum algorithms. Recently, provably efficient algorithms to solve this task for ground states within a gapped phase were developed. Here we dramatically improve the efficiency of these algorithms by showing how to learn properties of all ground states for systems with periodic boundary conditions from a single ground state sample. We prove that the prediction error tends to zero in the thermodynamic limit and numerically verify the results.

Figures (3)

• Figure 1: Summary of the algorithm. Given a representation of a single ground state of an equivariant Hamiltonian with long-range interactions, we would use its symmetries to think of different parts of the ground state as distinct training samples for a machine learning model trained for predicting a $p$-body observable. This way we can train a machine learning model for each distinct kind of a constituent term in a sum of $p$-body observables from a single ground state sample, and hence predict the sum itself for any other ground state within the topological phase.
• Figure 2: Classical shadows. Plotting an instance of all correlations $C_{ij}$ in a $32$ qubit Heisenberg chain, comparing the exact values above the diagonal to the predicted values below the diagonal.
• Figure 3: Numerical results.(a) and (c) Comparing the average root-mean-square error obtained with our method to the trivial predictor and the standard deviation of the testing samples, when predicting the ground state energy $O = H/\sqrt{n}$ as the system size increases. (b) The average RMS error obtained when predicting all $C_{ij} = \frac{1}{3}(X_i X_j + Y_i Y_j + Z_i Z_j)$ in the Heisenberg model using classical shadows, showing different distances of the two qubits separately, as the system size increases.